Lemma 10 (\(\mathsection \) 4.2 [1]}) 
Let \(G\)  have vertices \(v_1,\dots ,v_n\)  and let \(G^{\prime } = G-\lbrace v_n\rbrace \) . Pick \(m = (D(G^{\prime }), R^{\prime }, L^{\prime }) \in \mathcal {M}(G^{\prime })\)  uniformly at random. Let \(D_n(G) = \lbrace e_i : v_i \in N(v_n)\rbrace \) . For each \(e_i \in D_n(G)\) , let \(D_i(G) = D_i(G^{\prime }) \cup \lbrace d_i\rbrace \) . Let \(D(G) = \cup _{i=1}^n D_i(G)\) .
